Representing Waves as Sine and Cosine Functions - AP Physics 2

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Question

Which of the following electromagnetic waves has the highest frequency?

Answer

The frequency of a wave is directly related to the energy in the wave. The most energetic waves in the electromagnetic spectrum are gamma rays. Gamma rays are typically released from quasars, ultra-dense dying stars, and from atomic bomb blasts, which gives you the scope of the amount of energy involved with gamma rays.

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Question

Equation of sound wave 1:

$Asin(\omega_1t+\phi_1)$

Equation of sound wave 2:

$Asin(\omega_2t+\phi_2)$

Determine at what time there will be complete constructive interference between the two if:

$\omega_1=.5\frac{\pi}{s}$

$\omega_2=1\frac{\pi}{s}$

$\phi_2=\pi$

$\phi_1=\frac{\pi}{2}$

Answer

The will be destructive interference when

$Asin(\omega_1t+\phi_1)+Asin(\omega_2t+\phi_2)=0$

$Asin(\omega_1t+\phi_1)=-Asin(\omega_2t+\phi_2)$

$sin(\omega_1t+\phi_1)=-sin(\omega_2t+\phi_2)$

Using

$-sin(x)=sin(x+\pi)$

$sin(\omega_1t+\phi_1)=sin(\omega_2t+\phi_2+\pi)$

$\omega_1t+\phi_1=\omega_2t+\phi_2+\pi$

$\omega_1t-\omega_2t=\phi_2+\pi-\phi_1$

$(\omega_1-\omega_2)t=\phi_2+\pi-\phi_1$

$t=\frac{\phi_2+\pi-\phi_1}{\omega_1-\omega_2}$

Plugging in values:

$t=\frac{\frac{\pi}{2}+\pi-\pi}{\frac{\pi}{s}-\frac{.5\pi}{s}}$

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Question

For open pipes, the formula for wave patterns at any given time can be given by a Fourier Sine Series which is given as the infinite sum:

$y(x)=\sum_{n=0}^{\infty} sin(\frac{n\pi x}{l})$ when is an integer, and is the length of the pipe. Each individual value of is called a harmonic.

What is the angular frequency of the second harmonic?

Answer

Frequency, when given an equation in sine or cosine notation, is the value inside the parenthesis. For the second harmonic , the frequency is:

$\frac{2\pi }{l}$

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